Page 1 Representations, Inscriptions, Descriptions and Learning: A Kaleidoscope ofWindows 1
James J. KaputDepartment of MathematicsUniversity of Massachusetts at Dartmouth IntroductionMy task is to reflect on the papers in the two Special Issues on "Representations and thePsychology of Mathematics Education," Journal of Mathematical Behavior Vol. 17, Numbers 1and 2, especially to find common themes and opportunities for further progress. The papers wereproduced independently, but were based on and influenced by the discussions in the PMEWorking Group on Representations from 1990 to 1993. Gerald Goldin and Claude Janvier editedall the papers in the collection, and Goldin refers to several of them in his theoretical overviewpaper.The content of this paper is determined by the others mainly through complementarity and a focuson issues not directly addressed by, and perspectives not taken by, the papers. These issues andperspectives have to do with the roles of detailed features of conventional notations and studentnotational productions in problem solving and learning - how do notations actually work inparticular circumstances, especially in designed learning contexts? What do the decidedlycognitivist framework and language adopted in most of the papers prevent us from seeing orunderstanding? Finally, how can analyses of representational activity and the features of notationsinform the design of learning environments within computational media, especially those that fosterdeliberate and intense representational activity? I do not pretend to answer such questions, butrather to raise them in relation to the collected papers. I also offer an extended example to helpillustrate the beginnings of a notational analysis in the context of instructional design that might beconsidered complementary to the bulk of those provided in this collection.Authors use terms in different ways, and so I will discuss terminology at the outset in a way that isintended to frame and clarify usage across the papers. I shall begin with a discussion ofterminologies in relation to Goldin's goal of a "unified model" (Goldin, this issue), and then go onto point out how the same terms are used differently by Greer and Harel (1998), Even (1998),Cifarelli (this issue), Boulton-Lewis, Hall (1998), Owens & Clements (this issue), and Hitt (thisissue). Larer in the paper, a closer look at the paper by Mesquita (this issue) leads me to adiscussion of geometry in the computational medium and the impacts on the phenomena thatMesquita reports that result from a move to dynamic geometry. We also remark briefly on thestatus of mathematics as a language (Vergnaud, this issue).Of course, the particularity of language use is one window on authors’ assumptions, perspectivesand intent. I will use that window just as you will use it to understand me. So we are already atour first reflexive duality, looking at and looking through the language window whilesimultaneously being distracted by our own reflection in it. And what do we see through this
1
Work in this paper was supported by NSF Applications of Advanced Technology Program, Grant#RED 9619102 and Department of Education OERI grant # R305A60007. The views offered inthe paper are those of the author and need not reflect those of the Foundation or the Departmentof Education.----
Page 2warped (whorfed?) window? More windows, in a jumble, none straight or transparent, somereflecting into others, some translucent, some that seem to change when looked at, and some thatseem to be treated as writing surfaces. A peculiar, but inviting kaleidoscope.This area of study is notorious for its complexity and subtlety because it seems to connect toeverything we want to know or study. And our work is especially sensitive to foundationalassumptions about knowledge, mind, learning, language, development, and culture - assumptionsthat inevitably define ontologies, methodologies and explanatory objectives, sometimes explicitly,sometimes tacitly. It is also sensitive to point of view - to whether we use the language ofresearcher-observer, educator, or student. There is no neutral ground and no high ground fromwhich a privileged perspective is possible. But we must start somewhere, and we will begin withbasic issues of terminology and the assumptions and entailments of the ways the authors approachthe basic issues. We will then offer an illustration of the kinds of instructional design issues thatarise when representational activities occur in computational media.Terminologies and Goldin’s Goal of a “Unified Model”The Abstract Correspondence ApproachGoldin’s overview explicitly addresses terminology as he attempts a unified framework, which hedescribes as a “unified model.” It is less a model than a very general framework for a way oftalking about representational phenomena, problem solving, and (to a lesser extent) learning anddevelopment. In an aggressive effort to be broadly inclusive in his treatment, he begins where Idid (Kaput, 1985), with an abstract “correspondence” characterization of representation adaptedfrom Palmer (1977) that deliberately pays no attention to what kinds of things are involved in thecorrespondence - only that there be two entities that are taken, by an actor or an observer, to be insome referential relation to one another, one taken to “represent” the other. In the case of an actor,the referential connection may be experienced as a “standing for” or “corresponds to” relationbetween one part of her/his experience and another. Such a referential connection is hypothesizedby an observer to be expressed in actions (including writings and utterances). For an observer, thereferential connection may also include connections between an actor’s hypothesized mental eventsand externally observable actions on physical material. Goldin also asserts as representational suchcases of referential relationships as that between DNA and the biological material whose growth itcontrols. All such relationships, nonetheless, require an observer to be asserted into our collectiveworld. Following Palmer Goldin suggests, as did I in earlier work and in our joint work (Goldin& Kaput, 1992, 1996), that there follows an obligation to say what is representing what and inwhat ways.Such an abstract starting point enables us to talk about many different kinds of “representing” thatin languages other than English often have different designations, as many have pointed out. But italso requires us to distinguish these different kinds of representing. However, by its explicitnessregarding what is representing what and in what ways, it embodies biases towards a style ofdescription and assumptions regarding what is knowable and in what ways that may not beuniversally shared. Moreover, its generality may be misleading in the sense that other approachesand perspectives may not fit this style of description. I will illustrate with specific examplesshortly.It is probably not accidental that Goldin and I share a background in abstract mathematics, where apremium is put on generality, and where the operation of abstracting away from content-baseddetail is as natural as walking. However, I now feel that much is to be gained by adoptingalternative points of view while simultaneously exploiting the local conceptual stability of acorrespondence perspective.----
Page 3Problem Solving vs. Instructional DesignOne other important but tacit feature of Goldin’s approach is that it is born of a long-standing effortto understand problem solving independent of everyday classroom instruction. It is not rooted ininstructional design. This fact has large implications regarding its applicability. Indeed, thepapers, as a collection, are concerned more with problem solving rather than instruction. Ofcourse, most mathematics instructors attempt to teach problem solving skill, and most useproblems to teach mathematical content. A few of the papers take specific instructional sequencesto teach particular content as their object of study - the papers by Even (functions), Boulton-Lewis,and Hall (arithmetic using manipulatives). But none addresses issues of representation in thecontext of a specific extended curriculum. But, while none addresses issues of representation inthe context of a specific extended curriculum, this is the area where detailed analyses ofrepresentational strategies based on the work of the papers in this joint issue, coupled with longterm instructional design, may have the biggest practical payoffs. I will try to illustrate this pointvia an extended example in the last major section of the paper.Internal vs. External RepresentationsCognitivism and Dualism: Roads Not TakableGoldin, and in a more tacit way, most of the other authors make a fundamental distinction between“internal representations” and “external representations.” The former refer to hypothesized mentalconstructs and the latter to material notations of one kind or another. Several basiccognitivist/dualist assumptions are often, but not necessarily, wrapped up in this distinction,including the very idea of mental representation, which begs such questions as: What is it? Whatdo we mean when we say it “represents” something? For whom? How? What is the differencebetween the experience of an internal representation and that of an external representation? And isan external representation a socially or a personally constituted system? I should note that Goldin,however, characterizes representation sufficiently abstractly so as to avoid being trapped in strictdualist framework. His characterization is broad enough as a way of describing or accounting forobservations to include representations as descriptive of structures encoded physically in brains,neurons, DNA, etc.Aside from Vergnaud, the papers in this collection do not take up such questions directly, althoughmost work within this internal-external linguistic framework, with differing degrees ofexplicitness. Cifarrelli seems to use the word “representation” exclusively as mentalrepresentation, whereas Even uses the word to mean material representation. The other authorsmake the internal/external distinction at some level of explicitness, although the analysis byVergnaud complicates the distinction by attending to the actions from which mental structures areconstituted.To illustrate the challenges of encompassing all points of view from within a Goldin-likeframework, let us consider the phenomenon of fusion as examined by Nemirovsky & Monk (inpress). While we often discuss symbol and referent as if they are experienced as independententities, with some kind of specifiable connection between them - correspondence, association,indexical, etc. - Nemirovsky and Monk question whether this approach actually can account for thecreative functionality of symbol use in the lived-in world identified by Werner & Kaplan (1962),and where symbols are frequently not experientially distinguished from referents. They point outthat while identification of symbol and referent is treated by anthropologists as fetishism, animismand magic, and by psychologists as varieties of pathology, we treat regularly symbols in place ofwhat we know they stand for, despite the fact that we know that the picture of the person isdifferent from the person, or the drawn figure is not really a circle, the toy car is not a real car, thesquare we gestured in space with our hand is not a real square, etc. Nemirovsky and Monk (in----
Page 4press) note: “Fusion experiences are pervasive in everyday life and can adopt infinite forms, fromdiscussing directions on a map to commenting on a photograph; from drawing a face to gesturingthe shape of an object.” They go on to analyze in great detail the fusion experiences displayed in astudent’s conversations with a researcher about the construction and meanings of certain graphs ofmotion. The fusions involved the curve in the graph with the paths taken by the objects in motion,with the curvature shapes of the graph with the speed of the objects, and so on. Nemirovsky andMonk stress that fusion is a functional way of using symbols and tools, and not con-fusion. It is away of maintaining structure and orientation in time and in the space of actions and possibilitiessurrounding or activated by a symbol-rich experience.While Nemirovsky and Monk draw on the classic Werner-Kaplan (1962) approach tounderstanding symbol formation, I suggest that an evolutionary psychological perspective mightalso be fruitful. In particular, Donald (1991) examines the evolution of representational capacityfrom early primates to modern humans, and in doing so, he identifies a stage before spokenlanguage that he refers to as “mimetic” beginning about 1.5 million years ago and during whichsome major pre-human achievements occurred, including the use of fire for cooking, thedevelopment of sophisticated tools, migrations out of Africa, and so on. Mimetic culture involvedan intentional decentering, a use of the body (dance and gesture) to stand for or to refer tosomething else. Gesture, facial expression, body orientation and movement continue to play anessential role in the deep organization of experience and the tacit support of conversation. I suspectthat aspects of fusion have their roots in the mimetic dimension of human experience, althoughthey are expressed in symbolic behavior in our modern, symbol-saturated, linguistically mediatedculture.Speculations aside, however, we need to acknowledge that the kind of analysis offered byNemirovsky, Monk and others, provides insights to phenomena not easily reached from thecorrespondence view of representation and its dualist entailments. We will see further illustrationof the limits of this view below, although, for simplicity’s sake, we will continue to use thelanguage of the collection of papers to discuss them.Systems and Structures vs. Loose, Isolated Notational ElementsGoldin takes pains (as do Goldin & Kaput, 1992, 1996) to argue that “external representations”occur in systems. Such systems, while they can be personal and idiosyncratic under certaincircumstances, usually are cultural artifacts that cannot be separated from what is normally taken as“mathematical content.” Indeed, learning such systems and how to operate within them dominatesschool mathematics. Moreover, such systems are seldom used singly or in isolation from oneanother. Most mathematical activity involves multiple representation systems used in combinationwith one another, as Even illustrates so clearly. Several papers explicitly examine connectionsbetween external representation systems, including Boulton-Lewis, Hall, and Hitt. Greer andHarel deal with the question of how or whether students can learn to recognize common structuresacross different situations, situations that they would describe as “isomorphic.” Greer and Hareldo not, however, explicate these connections in terms of detailed notational and situationalparticulars.The Language Aspects of MathematicsBut what is such a thing as, say, the base-ten placeholder system? Is it internal or external, orneither? On one hand, it is a shared cultural artifact amounting to a language with referentialfunction and, most especially, through the very special organizations of actions upon it, a powerfulcomputational function. In another sense, it can appear concretely instantiated in a physicalmedium - paper, computer screen, whatever. But then, is it internal or external? And what does itrepresent (numbers?), for whom and under what circumstances? Is it a mathematical thing? Thatis, is it part of mathematics, or only a language used to represent and work with the realmathematical objects, whole numbers? One could ask similar questions about the Cartesian----
Page 5coordinate system. These kinds of questions arise when one begins with the internal/externaldistinction. On the other hand, as a way of framing discussion within a cognitivist perspective, itseems to have heuristic value.Vergnaud argues that it is incorrect to think of mathematics only as a language. The italicized“only” in the previous sentence reflects my interpretation of his point. My view is thatmathematics, as a means of organizing experience, rooted in schematically organizing action, mustinclude the expressive features of language. This expressive side of mathematics is invigoratedthrough the use of notation schemes instantiated within computational media (Shaffer & Kaput,submitted). While cognitive and social constructivist approaches to mathematics learning based onorganizing students’ actions into schema have provided credible outlines of instructional designregarding the objectsof mathematical languages, the challenge of how to build students’ expressivepower employing mathematical notation schemes remains. Somehow, the problem of how toorganize expressive acts in such a way as to produce expressive competence within conventionalnotation schemes and systems seems more difficult. For example, it seems easier to get a handleon students’ invention of algorithms within the number system notation system than it is to developmeans by which students can develop number systems. However, promising insights aredeveloping from various quarters, including, for example, diSessa, et al. (1991), Hoyles and Noss(1998), and the Freudenthal Institute (Gravemeijer, et al., in press). My hunch is that the basicnotation schemes are more fundamental and more complex cultural achievements than we mayrealize, and hence learning them as expressive tools requires more time and deeper expressiveengagement than we have devoted to date. The placeholder number system, the algebraic systems(including variables), and the coordinate systems are remarkable and hard-won culturalachievements within which most other mathematics is constituted.Inscriptions vs. Notation Schemes vs. Notation (or Representation) SystemsI suspect that I am one of a group who, in the early-mid 1980’s, introduced the phrase “multiplelinked representations.” As I noted in (Kaput, 1991), this perspective and language comesdangerously close to Platonism, and it is now clear that it is fundamentally cognitivist in spirit.The recent debate between cognitivists and situationists (Anderson, et. al, 1996; 1997; Greeno,1997; Cobb & Bowers, in press) has exposed the incommensurability of these theoretical stances.This incommensurability is starkly revealed in the lack of connection between the papers in thiscollection and the work in the situationist and activity-theoretic traditions. A forthcoming bookedited by Cobb, Yackel and McClain (in press), refers virtually not at all to the work described inthis collection, and vice-versa. These two bodies of work seem to exist in parallel universes.Another perspective on the representation problem is offered by certain researchers in the sociologyof scientific knowledge, led by Latour (1987, 1993) (see Roth & McGinn, 1998, for anintroductory review). For these researchers, the notion of inscription is central, an idea that I,influenced by Goodman (1976), approached via the construct of “representation scheme” in(Kaput, 1987) (I also used the phrase “notation scheme” interchangeably). One difference is thatthe word “inscription” is used by third-party observers to characterize marks in a physical mediumapart from any reference to how they might be used, understood, or perceived, and, apart from anystructure they might embody, from the third-party point of view. They are merely marks in amedium. “Representation scheme” was intended to refer to marks in a structured system, inprinciple machine-compilable, apart from their use in a representational way, standing forsomething else - similar to what Goldin and others in the collection refer to as an “externalrepresentation system.” The set of all finite linearly ordered sequences of numeric charactersserves as an example. “Notation system” (used interchangeably with “representation system” andeven “symbol system”) was intended to refer to a notation scheme used in a representational way.For example, the linearly ordered sequences of numeric characters used to represent permutationswould constitute a notation system. The same notation scheme could be used in a differentnotation system, e.g., to represent whole numbers rather than permutations. Put differently, the----
Page 6scheme does not include a referent, and the system does. (In both these cases we are ignoring theaction-structure that represents the composing of permutations and combining of numbers,respectively.) Since notation schemes are almost always discussed vis-à-vis their use to representsomething else, we almost always frame discussions in terms of systems rather than schemes.A second critical difference between inscription and notation scheme as theoretical constructs is thatthe former is used in a situationist explanatory framework and the latter in a cognitivist framework.Hence inscriptions are discussed relative to their production and active use, including how theyserve as “boundary objects” shared between discourse communities, how they are moved from oneplace to another, how they are overlaid, modified, discussed, and so on, in a social context apartfrom their possible connections with hypothetical cognitive events.The papers in this collection were conceived mostly before these ideas became current (which helpsexplain the lack of reference to their approaches and perspectives), and hence do not focus on thedetailed features of student produced inscriptions as they may bear upon the learning and problemsolving being studied. It should be noted that this style of analysis and the data on which it couldbe based are often not far from the analyses offered. Many papers include examples of studentinscriptions, e.g., Boulton-Lewis, Greer and Harel, Cifarelli, Even, Owens & Clements.However, we seldom see how the particular features of those inscriptions influence the course ofproblem solving and learning. We are all believe that they do, and the authors show that they do,but none showhow they do. This is a central question yet to be fully addressed.Addressing this question will necessarily require tracking how these features evolve as studentsprogress through problem solving or an instructional sequence. Cifarelli, near the end of hispaper, acknowledges the dynamic nature of student productions in problem solving. It would beinteresting to see this observation followed up in subsequent work, particularly in the style ofMeira (1992), for example, where microanalysis of student productions is at the heart of themethodology.Geometry in the Computational MediumThe papers in this collection, with the exception of those by Edwards, Hall and Even, focus on themathematics of static inert media. The papers by Even and Hall (and to a less direct extent thepaper by Hitt), focus on linked notations. However, certain papers point up explicit studentdifficulties with mathematics in static inert media that directly relate to the potentials of notations indynamic interactive media.Let us take a closer look at the paper by A. Lobo Mesquita which examines student difficulties withgeometric figures. In particular, she examines in detail the consequences of the fact that in mostschool geometry situations a drawn figure is, by necessity, a single concrete inscription whereas itis intended to stand for an idealized geometric object in what Poincare called the “GeometricSpace.” The physical drawing resides in what Poincare called the “Representative Space,” subjectto interpretation by human sensory apparatus. This subtle connection between the particular andthe general is understood only gradually by students as she illustrates.However, it is exactly this connection that is addressed by computer-based geometryenvironments, first the Geometric Supposers, and then dynamic geometry environments such asCabri and the Geometer’s Sketchpad. Here a particular construction (no longer merely a drawing),is subject to variation under the constraints of its construction. This has two immediate andpowerful consequences: (1) to expose the generality of the construction, since a given constructioncan usually have a continuum of instantiations as revealed by dragging any “free” point, and (2) thelogico-geometric structure of the construction is made more explicit through the patterns ofmovements of the figure as it is varied under the given constraints, with constructed andconsequent incidences, length-relations, symmetries, and so on, all preserved and subject to----
Page 7observation, inquiry, explanation and even proof. Since the examples and behaviors of the paperbear so directly on the two factors changed by dynamic geometry, it would be especially interestingif students’ reported reactions to the examples offered in the paper were compared with studentreactions to analogous figures constructed and manipulated in a dynamic geometry environment.We now turn to an illustration of how the computational medium offers notational opportunity forinstructional design within a curricular context.Representation and Instructional Design in Computational Media: A SampleAnalysis from CalculusTo illustrate the kinds of issues that arise when one approaches representational issues from aninstructional design perspective, I will provide an example interspersed with discussion of basicissues raised in the various papers.Anchoring Student Learning in Concrete, Experientially Real Data: TheInadequacy of the “Big Three” Linked RepresentationsHot, bi-directional links between pairs of the traditional “Big Three” (numerical, graphical andcharacter-string) notations have dominated the attention of educators and researchers (see the paperby Even), and, indeed, have driven computer software design and calculator design in recentyears. This has had the effect of enlarging the representational island and increasing students’ability to move around on it, but it has failed to connect to the students’ experiential mainland. Inthe words of Anna Sfard in her plenary address (with Patrick Thompson) at the PME-NA AnnualMeeting in October, 1994, “If the representations only represent each other, then the emperor isonly clothes.” A strong illustration of the kinds of learning that take place in the absence ofanchoring in concrete experience is given by Schoenfeld, et al. (1994). These researchersprovided an extremely detailed analysis of a single student’s learning in a linked “Big Three”environment, supplemented by a version of a target game based on Green Globs which involvesstudents defining functions whose graphs will pass through as many pre-given, randomlygenerated points on the plane as possible. The student’s learning was characterized by fragility,instability and disconnectedness from other knowledge or sense-making capability. (This was acapable, mature college-age student who had taken several prior courses involving algebra.) Thestudent’s conception of function was “only clothes” in the sense that, for her, the representationswere only referring to one another and not to any data associated with phenomena or situationsgrounded in her experience. By contrast, for the researchers the representations were physicalembodiments of their own ideas of function. Other reports of student learning difficulty in multiplelinked function environments indicate the inadequacy of linked representations and the strong needto provide experiential anchors for function representations.Our current work in the SimCalc Project (Kaput, Roschelle & Stroup, in press) puts phenomenaand situations at the center and treats the various representations of functions as means forunderstanding and reasoning about those phenomena and situations as reflected in Figure 1.----
Page 8Figure 1. Putting Phenomena at the CenterHere we are explicit about physical and cybernetic phenomena experienced as separate from theperson, but, perhaps generated in interaction with the person. Of course there are other kinds ofphenomena of direct interest, such as notational phenomena - the behavior of notations as weinteract with them. Or experienced kinesthetic phenomena, as occur in whole body motion orwhen a student moves an object using her hand.The reader will notice the bidirectionality of the arrows in Figure 1. The next two sections discussinterpretations of these arrows.From Representing to Creating and Controlling Situations or PhenomenaHistorically, we have assumed that mathematics was to be used to represent aspects of situations orphenomena within the notational systems of mathematics (often referred to as “modeling”) in orderto reason about and make sense of those situations or phenomena, which were taken as given.This can be taken as our intended meaning of the outward-pointing arrows. A secondinterpretation of the outward pointing arrows is data-transfer, from the physical environment to acomputational one through the use of measuring devices connected to a computational devicewhich can display the data in some mathematical notation system. Our work has focused on themathematics of motion, so in many cases, the notations describe velocities, positions, times, andcombinations of these in graphs, equations and tables, with our emphasis on graphs indicated inthe figure. However, such motion phenomena, and others such as fluid-flow, are not onlymodeled by the notations that describe them, they can be controlled by those notations. Thesephenomena can either be cybernetic, as with screen-objects whose movement is controlled by----
Page 9mathematical functions - represented as graphs, equations or tables - or physical, as with toy carslinked to a computer where their motion is controlled by mathematical functions defined on thecomputer. This can be taken as our intended meaning of the inward-pointing arrows. See(Nemirovsky, Kaput & Roschelle, 1998) for more details and concrete examples.These kinds of affordances turn a fundamental representational relationship between mathematicsand experience from one-way to bi-directional. This in turn supports a much tighter and morerapid interaction on which to base learning. Because the mathematical notation that controls aphenomenon also can be treated as a model of it, one can test a hypothetical model againstexpectations or predictions immediately - by “running” it. Note that the feedback structure oftenrequires twoFigure 2: Matching Red’s Position by Controlling Blue’s Velocityphenomena, P and P’, where P is given as a target and P’ is defined or controlled by the studentattempting to match P in another system of description. In cases involving rate-totals connections,P may act as a referent phenomenon for either a rate or totals description, and the student inputsone of these and gets feedback in terms of the other. For example, (See Figure 2) suppose we aregiven a vertical motion P of a Red elevator (on the left) and its position description (a position vs.time graph, for instance as in Figure 2). Then the student controls the motion P’ of a parallel Blueelevator (on Red’s right) by constructing its velocity, say a graph (or even a formula) to match themotion P of Red with feedback available by watching both elevators run simultaneously. Ofcentral importance is that the student’s intentions can be made visible, explicit and testable throughthe phenomena that the student controls. By exercising our many representational options, we canaddress an enormous range of learning objectives.From Multiple Linked Representations to Multiple Linked Descriptions ofExperientially Real Situations or PhenomenaFirst we set the stage for the distinction between different representations of the “same” descriptionand the “same” representation of “different” descriptions. Situations or phenomena admitting ofquantitative analysis almost always have two kinds of quantitative descriptions, one describing thetotal amount of the quantity at hand with respect to some other quantity such as time, and the other----
Page 10describing its rate of change with respect to that other quantity. In the SimCalc Project we havetaken the perspective that understanding the two-way relations between totals and rates descriptionsof varying quantities (and the situations that they describe), is a fundamental aspect of quantitativereasoning. It is exactly this relationship that is at the heart of the Fundamental Theorem ofCalculus, and indeed, at the heart of Calculus itself. Given the centrality of Calculus to ourcurriculum and to the mathematics, science and technology of western civilization, the connectionsbetween these two types of description is difficult to overestimate.Now, these rates-totals connections can be instantiated in any of the different representationsystems we have mentioned so far because both the rates and the totals descriptions are, in mostcases of interest to us, functions. In Figure 2 we saw a case where the two descriptions were bothinstantiated in the same representation system, coordinate graphs, but mentioned that they couldhave been represented across different systems. Furthermore, we continue to take advantage oflinked representations, so that we not only can connect graphs and formulas, we can cross-connect, for example, a rate graph to a totals formula. See Figure 3 for an illustration of the manypossible linkages.Figure 3: Linked Representations AND Linked DescriptionsSimilar points could be made about different descriptions in other mathematical domains, such asstatistical data analysis, probability, graph theory (especially rich representationally). In all thesecases, differences at the level of descriptions runs structurally deeper than do the differences inrepresentation systems, although it is of course the case that representations differ in their abilitiesto render that structure available to us. This description-representation issue is worthy of muchfurther study, especially cross-domain study.----
Page 11This new description-representation distinction adds another burden to our terminology and theneed to be explicit. To speak only of a “model” or a “representation” without further elaboration isclearly inadequate given the many different interpretations possible.The Invisible Effects of Static, Inert Media: Dominance of Character StringNotation SchemesFish don’t know they are wet. Similarly, we have historically taken without question the static,inert media in which notation schemes have been instantiated and used in representational andcomputational ways. And just as fish have gills and fins rather than lungs and legs, our ways ofdoing mathematics have certain features that evolved due to the nature of the media in which theyevolved. Thus, among notation systems, one type, the character string based systems, dominatesthe others. Character string systems are very compact, they can support intricate syntax for bothreading and transformations and so can support complex computation and reasoning far beyondwhat could be achieved by a notationally naked mind, and they can be representationally neutral inthe sense of being able to denote enormous varieties of referents without needing to share thevisual features of what they are representing (e.g., the meaning of “big” does not require the word“big” to be big).Changes from static inert to dynamic interactive medium affect the place of character string notationsystems in mathematics in at least three basic ways. The first is by enabling character strings to“come alive” - to embody procedures, or algorithms, that can be autonomously executed in order to“do something,” carry out a computation, evaluate an expression, perform a translation ortransformation, etc. - essentially to do anything a computer program can do. Second, the mediumcan support live linkages among notation systems. Third, it can support the instantiation of newnotation schemes that can be linked with others, new or old. We now turn attention to this third,most promising affordance.This deep notational bias is reflected in a deep cultural and curricular bias that has led to the bulk ofschool mathematics being centered on the teaching of reading and writing in character-basednotation schemes, especially routines for calculation or reasdoning. The momentum of thisnotational bias from static inert media led to the design of most first generation electroniceducational technology in mathematics to require character string input, e.g., keyboards andcalculator keys. This character string orientation extends to underlying issues of legitimacy - whatcounts as significant mathematics is typically taken to be mathematics expressed in terms ofcharacter strings. While we have no reason to expect that the extraordinarily powerful of characterstring mathematics will ever or should ever disappear or be displaced, we should expect that it willbe increasingly augmented by mathematics expressed in other notational styles, especially thosethat draw upon the new structurable flexibilities and visually rich notations of dynamic, graphicand direct-manipulation media. We see in the papers by Edwards, Even and Hitt (this issue),gradual movement away from exclusive dependence on character strings. Of course, the work byJanvier beginning in the 1970’s, pioneered the study of graphical notation systems. Hisposthumously published paper in this issue raises the question of how the subtle and pervasivesemantics of time structures descriptions of phenomena, whether or not the key variables used todescribe the phenomena are themselves temporal.Historical Approaches: Globally Defined Functions Are PrimaryHistorical necessity pushed mathematics towards globally defined functions - a character stringdefinition of a function or quantitative relationship was globally defined (over its natural domain)essentially by default - through the identification of the function or quantitative relationship with thecharacter string. But, of course, the phenomena and situations that these are used to model do notusually have such “simply” definable characteristics over unbounded domains. Rather, they are----
Page 12defined over finite extent (domains) and often change their essential character within that finiteextent. Much elaborate and beautiful mathematics has been produced to achieve the flexibility thatclosed form functions do not admit - infinite series approximations of various kinds, Fourierseries, and so on - all of which require an apprenticeship in the algebraic world that most citizenstoday cannot afford and whose role in sense-making is fundamentally transformed by newtechnologies. (Note that their significance does not disappear, but is played out in different ways,e.g., by being embodied in tools that are used by people who may not need to master that style ofmathematics.)A Alternative: Piecewise Defined, Visually Editable FunctionsIn MathWorlds, an important feature is the ability to construct and modify functions that areconnected to motion and other simulations through strictly graphical means - direct manipulation ofpiecewise defined functions (see also Yerushalmy, 1997). As indicated in Figure 4 below, one cancreate a piecewise constant velocity function that controls the motion of the Clown in the motionsimulation by dragging the black dots associated with parts of the graphs.Drag this hotspotleft or right toadjust the initialtimeDrag this hotspotleft or right toadjust the initialtimeDrag this hotspotleft or right toadjust the initialtimeFigure 4: Visually Editable Velocity GraphsAn important point is that the historical power of algebra lay in its status as an action notationscheme, one that supports structured actions on well-formed character strings in support ofmathematical reasoning and modeling. By contrast, in static inert media, coordinate graphs havetypically been used as static notations - to be inspected, and analyzed as fixed inscriptions. Inrecent times, by instantiating coordinate graphs in dynamic interactive media as linked to algebraicrepresentations, as noted above they become manipulable, but in global form. That is, one actsuniformly on the function over its entire domain - translating, stretching, reflecting, etc. (SeeConfrey, 1991) However, the approach illustrated in Figure 4 goes a step beyond this, in notrequiring an algebraic interpretation or linkage. Instead, much in the spirit of Dynamic Geometry,the graphs are directly manipulable: pieces can be inserted and modified, etc. The manipulability ofalgebraic notation has been replaced by a new kind of manipulabilty.----
Page 13Three sets of reasons lie behind our use in the SimCalc Project of piecewise defined functions: (1)from a modeling orientation, phenomena occur in segments of time, each of which has boundedextent, so the phenomenon usually is not regarded as being defined in a unitary fashion “forever;”(2) work by Nemirovsky and colleagues has established that people naturally tend to interpretgraphs on the basis of interval-analyses (Nemirovsky, 1994, 1996; Nemirovsky, Tierney &Wright, in press; Tierney & Nemirovsky, 1995; Monk & Nemirovsky, 1994), whereincomplicated graphs linked to phenomena are parsed one section or interval at a time according tothe student’s current experience of both the graph and whatever it is linked to; and (3) in order toachieve the variation needed to avoid the logical and psychological degeneracy of constant or linearfunctions while preserving computational tractability, we chose to begin with graphically editablepiecewise constant and linear functions whose slope and area computations can be approachedusing simple arithmetic and geometry, temporarily avoiding the subtleties of limits andapproximations (Kaput, Roschelle & Stroup, in press).Curricular Implications: Moving Past the Algebra BottleneckIn addition to the factors identified above, the weight of the historical tradition based in theextraordinary achievements of algebra-based mathematics in the hands of the masters who inventedand used it (these constitute some of the greatest intellectual achievements of Western civilization),the assumption that the mathematics of change and variation, including the ideas and techniques ofcalculus, should be learned exclusively in the language of algebra - this assumption - hascontinued by default up to the current time. It is deeply embedded in our curriculum and ourinstitutional structures (Kaput, 1997), and indeed, it is reflected in both our popular and intellectualcultures.No one should presume to challenge the power of algebra, and indeed, much mathematicsabsolutely requires algebra, including most of classical mathematics. Rather, we can and shouldchallenge its currently dominant place in the curriculum as a prerequisite for access to otherimportant mathematics by students who will never need the specialized techniques that the algebramakes possible. On the other hand, all that we have learned about student learning suggests thatstudents bring enormously rich resources to us that are intimately tied to their natural ways of beingin and constructing their world. Our ongoing work provides strong evidence that these resources,combined with technologically supported learning environments that instantiate new notationsystems and ways of acting upon them, offer dramatically new possibilities for mathematicslearning.ConclusionsWe are entering a new era in the study of representation, where new issues need attention and newperspectives and analytical tools are becoming available. The same factors that are being felt acrossour society and our intellectual landscape are at work here: new technologies yield new media thatalter the semiotic foundations of mathematics; situationist perspectives provide new ways tointegrate, or supersede, traditional cognitivist and social-psychological perspective and methods;advances in the sociology of science provide fresh perspectives on the creation and communicationof the inscriptions that move among us and that constitute an essential dimension of our world; andnew obligations are opposed upon mathematics educators to render ever more mathematicslearnable by ever more, and hence more diverse, people.We are entering a new era in the study of representation, where new issues need attention and newperspectives and analytical tools are becoming available. The same factors that are being felt acrossour society and our intellectual landscape are at work here: new technologies yield new media thatalter the semiotic foundations of mathematics; situationist perspectives provide new ways tointegrate, or supercede, traditional cognitivist and social-psychological perspectives and methods;advances in the sociology of science provide fresh perspectives on the creation and communication----
Page 14of the inscriptions that move among us and constitute an essential dimension of our world; andnew obligations are imposed upon mathematics educators to render ever more mathematicslearnable by ever more, and hence more diverse, people.I hope that this paper, as a small twist of the representational kaleidoscope, has provided a fewindications of how the work in this collection can serve the needs of the new era. Of specialinterest is how we may extend this work to inform the design of instruction, especially in thecontexts of school-implementable curricula, which is where we ultimately turn for a fair test of anyof our endeavors.----
Page 15ReferencesAnderson, J. R., Reder, L. M., & Simon, H. A. (1996). Situated learning and education.Educational Researcher,25(4), 5-11.Anderson, J. R., Reder, L. M., & Simon, H. A. (1997). Situative vs. cognitive perspective:Form vs. substance. Educational Researcher,26(1), 18-21.Cobb, P., & Bowers, J. (in press). Cognitive and situated learning perspectives in theory andpractice. To appear in Educational Researcher.Cobb, P., Yackel, E., & McClain, K. (in press). Symbolizing and communicating inmathematics classrooms. Hillsdale, NJ: Erlbaum.Confrey, J., & Smith, E. (1991). A framework for functions: Prototypes, multiplerepresentations, and transformations. Paper presented at the Proceedings of the thirteenth annualmeeting of Psychology of Mathematics Education-NA, Blacksburg, VA.diSessa, A. A., Hammer, D., Sherin, B. & Kolpakowski, T. (1991). Inventing graphing: Meta-representational expertise in children. Journal of Mathematical Behavior, 10, 117-160.Donald, M. (1991). Origins of the modern mind: Three stages in the evolution of culture andcognition. Cambridge, MA: Harvard University Press.Goodman, N. (1976). Languages of art. (Revised ed.). Amherst, MA: University ofMassachusetts Press.Goldin, G. A., & Kaput, J. (1992). Comments at Working Group on Representations, AnnualMeeting of the International Group for the Psychology of Mathematics Education, Durham, NewHampshire.Gravemeijer, K., Cobb, P., Bowers, J., & Whitenack, J. (in press). Symbolizing, modeling,and instructional design. To appear in P. Cobb, E. Yackel, and K. McClain (Eds.), Symbolizingand communicating in mathematics classrooms. Hillsdale, NJ: Erlbaum.Greeno, J. (1997). On claims that answer the wrong questions. Educational Researcher, 26(1),5-17.Hoyles, C., & Noss, R. (Eds.) (1998). Mathematics for a new millennium. London, England:Springer-Verlag.Kaput, J. (1985). Representation and problem solving: Methodological issues related tomodeling. In E. Silver (Ed.), Teaching and learning mathematical problem solving: Multipleresearch perspectives, (pp. 381-398). Hillsdale, NJ: Lawrence Erlbaum.Kaput, J. (1987). Toward a theory of symbol use in mathematics. In C. Janvier (Ed.), Problemsof representation in mathematics learning and problem solving, (pp. 159-196). Hillsdale, NJ:Lawrence Erlbaum Associates.Kaput, J. (1991). Notations and representations as mediators of constructive processes. In E. v.Glasersfeld (Ed.), Constructivism and mathematics education, (pp. 53-74). Dordrecht,Netherlands: Kluwer.----
Page 16Kaput, J. (1997). Rethinking calculus in terms of learning and thinking. The AmericanMathematics Monthly, 104 (8), 731-737.Kaput, J., Roschelle, J., & Stroup, W. (in press). SimCalc: Accelerating Students' Engagementwith the Math of Change. To appear in M. Jacobson & R. Kozma (Eds.) Learning the sciences ofthe 21
st
century: Research, design, and implementation of advanced technology learningenvironments. Hillsdale, NJ: Lawrence Erlbaum.Latour, B. (1987). Science in action: How to follow scientists and engineers through society.Cambridge, MA: Harvard University Press.Latour, B. (1993). La clef de Berlin et autres lecons d’un amateur de sciences [The key to Berlinand other lessons of a science lover]. Paris: Editions la Decouverte.Meira, L. (1992). The microevolution of mathematical representations in children's activity.Paper presented at the Proceedings of the Sixteenth Annual Conference of the PME, Vol. 2,Durham, NH.Monk, S., & Nemirovsky, R. (1994) The Case of Dan: Student construction of a functionalsituation through visual attributes. In E. Dubinsky, J. Kaput, & A. Schoenfeld (Eds.), Researchin Collegiate Mathematics Education. Vol. 1. Providence, RI: American Mathematics Society,139-168.Nemirovsky, R. (1994). On ways of symbolizing: The case of Laura and velocity sign. TheJournal of Mathematical Behavior, 13, 389-422Nemirovsky, R. (1996). Mathematical narratives. In N. Bednarz, C. Kieran, & L. Lee (Eds.),Approaches to algebra: Perspectives for research and teaching, (p. 197-223). Dordrecht, TheNetherlands: Kluwer Academic Publishers.Nemirovsky, R., Kaput, J., & Roschelle, J. (1998). Enlarging mathematical activity frommodeling phenomena to generating phenomena. Paper presented at the Proceedings of the 22
nd
International Conference of the Psychology of Mathematics Education in Stellenbosch, SouthAfrica.Nemirovsky, R., & Monk, S. (in press). “If you look at it the other way...” An exploration intothe nature of symbolizing. In Cobb, P., Yackel, E. and McClain, K. (Eds.) Symbolizing andcommunicating in mathematics classrooms. Hillsdale, NJ: Erlbaum.Nemirovsky, R., Tierney, C., Wright, T. (in press). Body motion and graphing. Cognition andInstruction.Palmer, S. E. (1977). Fundamental aspects of cognitive representation. In E. Rosch & B. B.Lloyd (Eds.), Cognition and categorization. Hillsdale, NJ: Lawrence Erlbaum Associates.Roth, W-M., & McGinn, M. K. (1998). Inscriptions: Toward a theory of representing as socialpractice. Review of Education Research, 68(1), 35-59.Schoenfeld, A., Smith, J., & Arcavi, A. (1994). Learning: The microgenetic analysis of onestudent’s evolving understanding of a complex subject matter domain. In R. Glaser (Ed.),Advances in instructional psychology, (Vol. 4). Hillsdale, NJ: Lawrence Erlbaum.Shaffer, D., & Kaput, J. (submitted). Mathematics and virtual culture: An evolutionaryperspective on technology and mathematics education. Educational Studies in Mathematics.----
Page 17Tierney, C., & Nemirovsky, R. (1995). Children's Graphing of Changing Situations. Presentedat the 1995 Annual Meeting of the American Educational Research Association. San Francisco,CA.Werner, H., & Kaplan, B. (1962). Symbol formation. New York: Wiley.Yerushalmy, M. (1997). Emergence of new schemes for solving algebra word problems: Theimpact of technology and the function approach. Paper presented at the Proceedings of the 21
st
Conference of the International Group for the Psychology of Mathematics Education, Lahti,Finland.
.
Google automatically generates html versions of documents as we crawl the web.
Page 1
Representations, Inscriptions, Descriptions and Learning: A Kaleidoscope ofWindows
1
James J. KaputDepartment of MathematicsUniversity of Massachusetts at Dartmouth
IntroductionMy task is to reflect on the papers in the two Special Issues on "Representations and thePsychology of Mathematics Education," Journal of Mathematical Behavior Vol. 17, Numbers 1and 2, especially to find common themes and opportunities for further progress. The papers wereproduced independently, but were based on and influenced by the discussions in the PMEWorking Group on Representations from 1990 to 1993. Gerald Goldin and Claude Janvier editedall the papers in the collection, and Goldin refers to several of them in his theoretical overviewpaper.The content of this paper is determined by the others mainly through complementarity and a focuson issues not directly addressed by, and perspectives not taken by, the papers. These issues andperspectives have to do with the roles of detailed features of conventional notations and studentnotational productions in problem solving and learning - how do notations actually work inparticular circumstances, especially in designed learning contexts? What do the decidedlycognitivist framework and language adopted in most of the papers prevent us from seeing orunderstanding? Finally, how can analyses of representational activity and the features of notationsinform the design of learning environments within computational media, especially those that fosterdeliberate and intense representational activity? I do not pretend to answer such questions, butrather to raise them in relation to the collected papers. I also offer an extended example to helpillustrate the beginnings of a notational analysis in the context of instructional design that might beconsidered complementary to the bulk of those provided in this collection.Authors use terms in different ways, and so I will discuss terminology at the outset in a way that isintended to frame and clarify usage across the papers. I shall begin with a discussion ofterminologies in relation to Goldin's goal of a "unified model" (Goldin, this issue), and then go onto point out how the same terms are used differently by Greer and Harel (1998), Even (1998),Cifarelli (this issue), Boulton-Lewis, Hall (1998), Owens & Clements (this issue), and Hitt (thisissue). Larer in the paper, a closer look at the paper by Mesquita (this issue) leads me to adiscussion of geometry in the computational medium and the impacts on the phenomena thatMesquita reports that result from a move to dynamic geometry. We also remark briefly on thestatus of mathematics as a language (Vergnaud, this issue).Of course, the particularity of language use is one window on authors’ assumptions, perspectivesand intent. I will use that window just as you will use it to understand me. So we are already atour first reflexive duality, looking at and looking through the language window whilesimultaneously being distracted by our own reflection in it. And what do we see through this
1
Work in this paper was supported by NSF Applications of Advanced Technology Program, Grant#RED 9619102 and Department of Education OERI grant # R305A60007. The views offered inthe paper are those of the author and need not reflect those of the Foundation or the Departmentof Education.----
Page 2warped (whorfed?) window? More windows, in a jumble, none straight or transparent, somereflecting into others, some translucent, some that seem to change when looked at, and some thatseem to be treated as writing surfaces. A peculiar, but inviting kaleidoscope.This area of study is notorious for its complexity and subtlety because it seems to connect toeverything we want to know or study. And our work is especially sensitive to foundationalassumptions about knowledge, mind, learning, language, development, and culture - assumptionsthat inevitably define ontologies, methodologies and explanatory objectives, sometimes explicitly,sometimes tacitly. It is also sensitive to point of view - to whether we use the language ofresearcher-observer, educator, or student. There is no neutral ground and no high ground fromwhich a privileged perspective is possible. But we must start somewhere, and we will begin withbasic issues of terminology and the assumptions and entailments of the ways the authors approachthe basic issues. We will then offer an illustration of the kinds of instructional design issues thatarise when representational activities occur in computational media.Terminologies and Goldin’s Goal of a “Unified Model”The Abstract Correspondence ApproachGoldin’s overview explicitly addresses terminology as he attempts a unified framework, which hedescribes as a “unified model.” It is less a model than a very general framework for a way oftalking about representational phenomena, problem solving, and (to a lesser extent) learning anddevelopment. In an aggressive effort to be broadly inclusive in his treatment, he begins where Idid (Kaput, 1985), with an abstract “correspondence” characterization of representation adaptedfrom Palmer (1977) that deliberately pays no attention to what kinds of things are involved in thecorrespondence - only that there be two entities that are taken, by an actor or an observer, to be insome referential relation to one another, one taken to “represent” the other. In the case of an actor,the referential connection may be experienced as a “standing for” or “corresponds to” relationbetween one part of her/his experience and another. Such a referential connection is hypothesizedby an observer to be expressed in actions (including writings and utterances). For an observer, thereferential connection may also include connections between an actor’s hypothesized mental eventsand externally observable actions on physical material. Goldin also asserts as representational suchcases of referential relationships as that between DNA and the biological material whose growth itcontrols. All such relationships, nonetheless, require an observer to be asserted into our collectiveworld. Following Palmer Goldin suggests, as did I in earlier work and in our joint work (Goldin& Kaput, 1992, 1996), that there follows an obligation to say what is representing what and inwhat ways.Such an abstract starting point enables us to talk about many different kinds of “representing” thatin languages other than English often have different designations, as many have pointed out. But italso requires us to distinguish these different kinds of representing. However, by its explicitnessregarding what is representing what and in what ways, it embodies biases towards a style ofdescription and assumptions regarding what is knowable and in what ways that may not beuniversally shared. Moreover, its generality may be misleading in the sense that other approachesand perspectives may not fit this style of description. I will illustrate with specific examplesshortly.It is probably not accidental that Goldin and I share a background in abstract mathematics, where apremium is put on generality, and where the operation of abstracting away from content-baseddetail is as natural as walking. However, I now feel that much is to be gained by adoptingalternative points of view while simultaneously exploiting the local conceptual stability of acorrespondence perspective.----
Page 3Problem Solving vs. Instructional DesignOne other important but tacit feature of Goldin’s approach is that it is born of a long-standing effortto understand problem solving independent of everyday classroom instruction. It is not rooted ininstructional design. This fact has large implications regarding its applicability. Indeed, thepapers, as a collection, are concerned more with problem solving rather than instruction. Ofcourse, most mathematics instructors attempt to teach problem solving skill, and most useproblems to teach mathematical content. A few of the papers take specific instructional sequencesto teach particular content as their object of study - the papers by Even (functions), Boulton-Lewis,and Hall (arithmetic using manipulatives). But none addresses issues of representation in thecontext of a specific extended curriculum. But, while none addresses issues of representation inthe context of a specific extended curriculum, this is the area where detailed analyses ofrepresentational strategies based on the work of the papers in this joint issue, coupled with longterm instructional design, may have the biggest practical payoffs. I will try to illustrate this pointvia an extended example in the last major section of the paper.Internal vs. External RepresentationsCognitivism and Dualism: Roads Not TakableGoldin, and in a more tacit way, most of the other authors make a fundamental distinction between“internal representations” and “external representations.” The former refer to hypothesized mentalconstructs and the latter to material notations of one kind or another. Several basiccognitivist/dualist assumptions are often, but not necessarily, wrapped up in this distinction,including the very idea of mental representation, which begs such questions as: What is it? Whatdo we mean when we say it “represents” something? For whom? How? What is the differencebetween the experience of an internal representation and that of an external representation? And isan external representation a socially or a personally constituted system? I should note that Goldin,however, characterizes representation sufficiently abstractly so as to avoid being trapped in strictdualist framework. His characterization is broad enough as a way of describing or accounting forobservations to include representations as descriptive of structures encoded physically in brains,neurons, DNA, etc.Aside from Vergnaud, the papers in this collection do not take up such questions directly, althoughmost work within this internal-external linguistic framework, with differing degrees ofexplicitness. Cifarrelli seems to use the word “representation” exclusively as mentalrepresentation, whereas Even uses the word to mean material representation. The other authorsmake the internal/external distinction at some level of explicitness, although the analysis byVergnaud complicates the distinction by attending to the actions from which mental structures areconstituted.To illustrate the challenges of encompassing all points of view from within a Goldin-likeframework, let us consider the phenomenon of fusion as examined by Nemirovsky & Monk (inpress). While we often discuss symbol and referent as if they are experienced as independententities, with some kind of specifiable connection between them - correspondence, association,indexical, etc. - Nemirovsky and Monk question whether this approach actually can account for thecreative functionality of symbol use in the lived-in world identified by Werner & Kaplan (1962),and where symbols are frequently not experientially distinguished from referents. They point outthat while identification of symbol and referent is treated by anthropologists as fetishism, animismand magic, and by psychologists as varieties of pathology, we treat regularly symbols in place ofwhat we know they stand for, despite the fact that we know that the picture of the person isdifferent from the person, or the drawn figure is not really a circle, the toy car is not a real car, thesquare we gestured in space with our hand is not a real square, etc. Nemirovsky and Monk (in----
Page 4press) note: “Fusion experiences are pervasive in everyday life and can adopt infinite forms, fromdiscussing directions on a map to commenting on a photograph; from drawing a face to gesturingthe shape of an object.” They go on to analyze in great detail the fusion experiences displayed in astudent’s conversations with a researcher about the construction and meanings of certain graphs ofmotion. The fusions involved the curve in the graph with the paths taken by the objects in motion,with the curvature shapes of the graph with the speed of the objects, and so on. Nemirovsky andMonk stress that fusion is a functional way of using symbols and tools, and not con-fusion. It is away of maintaining structure and orientation in time and in the space of actions and possibilitiessurrounding or activated by a symbol-rich experience.While Nemirovsky and Monk draw on the classic Werner-Kaplan (1962) approach tounderstanding symbol formation, I suggest that an evolutionary psychological perspective mightalso be fruitful. In particular, Donald (1991) examines the evolution of representational capacityfrom early primates to modern humans, and in doing so, he identifies a stage before spokenlanguage that he refers to as “mimetic” beginning about 1.5 million years ago and during whichsome major pre-human achievements occurred, including the use of fire for cooking, thedevelopment of sophisticated tools, migrations out of Africa, and so on. Mimetic culture involvedan intentional decentering, a use of the body (dance and gesture) to stand for or to refer tosomething else. Gesture, facial expression, body orientation and movement continue to play anessential role in the deep organization of experience and the tacit support of conversation. I suspectthat aspects of fusion have their roots in the mimetic dimension of human experience, althoughthey are expressed in symbolic behavior in our modern, symbol-saturated, linguistically mediatedculture.Speculations aside, however, we need to acknowledge that the kind of analysis offered byNemirovsky, Monk and others, provides insights to phenomena not easily reached from thecorrespondence view of representation and its dualist entailments. We will see further illustrationof the limits of this view below, although, for simplicity’s sake, we will continue to use thelanguage of the collection of papers to discuss them.Systems and Structures vs. Loose, Isolated Notational ElementsGoldin takes pains (as do Goldin & Kaput, 1992, 1996) to argue that “external representations”occur in systems. Such systems, while they can be personal and idiosyncratic under certaincircumstances, usually are cultural artifacts that cannot be separated from what is normally taken as“mathematical content.” Indeed, learning such systems and how to operate within them dominatesschool mathematics. Moreover, such systems are seldom used singly or in isolation from oneanother. Most mathematical activity involves multiple representation systems used in combinationwith one another, as Even illustrates so clearly. Several papers explicitly examine connectionsbetween external representation systems, including Boulton-Lewis, Hall, and Hitt. Greer andHarel deal with the question of how or whether students can learn to recognize common structuresacross different situations, situations that they would describe as “isomorphic.” Greer and Hareldo not, however, explicate these connections in terms of detailed notational and situationalparticulars.The Language Aspects of MathematicsBut what is such a thing as, say, the base-ten placeholder system? Is it internal or external, orneither? On one hand, it is a shared cultural artifact amounting to a language with referentialfunction and, most especially, through the very special organizations of actions upon it, a powerfulcomputational function. In another sense, it can appear concretely instantiated in a physicalmedium - paper, computer screen, whatever. But then, is it internal or external? And what does itrepresent (numbers?), for whom and under what circumstances? Is it a mathematical thing? Thatis, is it part of mathematics, or only a language used to represent and work with the realmathematical objects, whole numbers? One could ask similar questions about the Cartesian----
Page 5coordinate system. These kinds of questions arise when one begins with the internal/externaldistinction. On the other hand, as a way of framing discussion within a cognitivist perspective, itseems to have heuristic value.Vergnaud argues that it is incorrect to think of mathematics only as a language. The italicized“only” in the previous sentence reflects my interpretation of his point. My view is thatmathematics, as a means of organizing experience, rooted in schematically organizing action, mustinclude the expressive features of language. This expressive side of mathematics is invigoratedthrough the use of notation schemes instantiated within computational media (Shaffer & Kaput,submitted). While cognitive and social constructivist approaches to mathematics learning based onorganizing students’ actions into schema have provided credible outlines of instructional designregarding the objectsof mathematical languages, the challenge of how to build students’ expressivepower employing mathematical notation schemes remains. Somehow, the problem of how toorganize expressive acts in such a way as to produce expressive competence within conventionalnotation schemes and systems seems more difficult. For example, it seems easier to get a handleon students’ invention of algorithms within the number system notation system than it is to developmeans by which students can develop number systems. However, promising insights aredeveloping from various quarters, including, for example, diSessa, et al. (1991), Hoyles and Noss(1998), and the Freudenthal Institute (Gravemeijer, et al., in press). My hunch is that the basicnotation schemes are more fundamental and more complex cultural achievements than we mayrealize, and hence learning them as expressive tools requires more time and deeper expressiveengagement than we have devoted to date. The placeholder number system, the algebraic systems(including variables), and the coordinate systems are remarkable and hard-won culturalachievements within which most other mathematics is constituted.Inscriptions vs. Notation Schemes vs. Notation (or Representation) SystemsI suspect that I am one of a group who, in the early-mid 1980’s, introduced the phrase “multiplelinked representations.” As I noted in (Kaput, 1991), this perspective and language comesdangerously close to Platonism, and it is now clear that it is fundamentally cognitivist in spirit.The recent debate between cognitivists and situationists (Anderson, et. al, 1996; 1997; Greeno,1997; Cobb & Bowers, in press) has exposed the incommensurability of these theoretical stances.This incommensurability is starkly revealed in the lack of connection between the papers in thiscollection and the work in the situationist and activity-theoretic traditions. A forthcoming bookedited by Cobb, Yackel and McClain (in press), refers virtually not at all to the work described inthis collection, and vice-versa. These two bodies of work seem to exist in parallel universes.Another perspective on the representation problem is offered by certain researchers in the sociologyof scientific knowledge, led by Latour (1987, 1993) (see Roth & McGinn, 1998, for anintroductory review). For these researchers, the notion of inscription is central, an idea that I,influenced by Goodman (1976), approached via the construct of “representation scheme” in(Kaput, 1987) (I also used the phrase “notation scheme” interchangeably). One difference is thatthe word “inscription” is used by third-party observers to characterize marks in a physical mediumapart from any reference to how they might be used, understood, or perceived, and, apart from anystructure they might embody, from the third-party point of view. They are merely marks in amedium. “Representation scheme” was intended to refer to marks in a structured system, inprinciple machine-compilable, apart from their use in a representational way, standing forsomething else - similar to what Goldin and others in the collection refer to as an “externalrepresentation system.” The set of all finite linearly ordered sequences of numeric charactersserves as an example. “Notation system” (used interchangeably with “representation system” andeven “symbol system”) was intended to refer to a notation scheme used in a representational way.For example, the linearly ordered sequences of numeric characters used to represent permutationswould constitute a notation system. The same notation scheme could be used in a differentnotation system, e.g., to represent whole numbers rather than permutations. Put differently, the----
Page 6scheme does not include a referent, and the system does. (In both these cases we are ignoring theaction-structure that represents the composing of permutations and combining of numbers,respectively.) Since notation schemes are almost always discussed vis-à-vis their use to representsomething else, we almost always frame discussions in terms of systems rather than schemes.A second critical difference between inscription and notation scheme as theoretical constructs is thatthe former is used in a situationist explanatory framework and the latter in a cognitivist framework.Hence inscriptions are discussed relative to their production and active use, including how theyserve as “boundary objects” shared between discourse communities, how they are moved from oneplace to another, how they are overlaid, modified, discussed, and so on, in a social context apartfrom their possible connections with hypothetical cognitive events.The papers in this collection were conceived mostly before these ideas became current (which helpsexplain the lack of reference to their approaches and perspectives), and hence do not focus on thedetailed features of student produced inscriptions as they may bear upon the learning and problemsolving being studied. It should be noted that this style of analysis and the data on which it couldbe based are often not far from the analyses offered. Many papers include examples of studentinscriptions, e.g., Boulton-Lewis, Greer and Harel, Cifarelli, Even, Owens & Clements.However, we seldom see how the particular features of those inscriptions influence the course ofproblem solving and learning. We are all believe that they do, and the authors show that they do,but none showhow they do. This is a central question yet to be fully addressed.Addressing this question will necessarily require tracking how these features evolve as studentsprogress through problem solving or an instructional sequence. Cifarelli, near the end of hispaper, acknowledges the dynamic nature of student productions in problem solving. It would beinteresting to see this observation followed up in subsequent work, particularly in the style ofMeira (1992), for example, where microanalysis of student productions is at the heart of themethodology.Geometry in the Computational MediumThe papers in this collection, with the exception of those by Edwards, Hall and Even, focus on themathematics of static inert media. The papers by Even and Hall (and to a less direct extent thepaper by Hitt), focus on linked notations. However, certain papers point up explicit studentdifficulties with mathematics in static inert media that directly relate to the potentials of notations indynamic interactive media.Let us take a closer look at the paper by A. Lobo Mesquita which examines student difficulties withgeometric figures. In particular, she examines in detail the consequences of the fact that in mostschool geometry situations a drawn figure is, by necessity, a single concrete inscription whereas itis intended to stand for an idealized geometric object in what Poincare called the “GeometricSpace.” The physical drawing resides in what Poincare called the “Representative Space,” subjectto interpretation by human sensory apparatus. This subtle connection between the particular andthe general is understood only gradually by students as she illustrates.However, it is exactly this connection that is addressed by computer-based geometryenvironments, first the Geometric Supposers, and then dynamic geometry environments such asCabri and the Geometer’s Sketchpad. Here a particular construction (no longer merely a drawing),is subject to variation under the constraints of its construction. This has two immediate andpowerful consequences: (1) to expose the generality of the construction, since a given constructioncan usually have a continuum of instantiations as revealed by dragging any “free” point, and (2) thelogico-geometric structure of the construction is made more explicit through the patterns ofmovements of the figure as it is varied under the given constraints, with constructed andconsequent incidences, length-relations, symmetries, and so on, all preserved and subject to----
Page 7observation, inquiry, explanation and even proof. Since the examples and behaviors of the paperbear so directly on the two factors changed by dynamic geometry, it would be especially interestingif students’ reported reactions to the examples offered in the paper were compared with studentreactions to analogous figures constructed and manipulated in a dynamic geometry environment.We now turn to an illustration of how the computational medium offers notational opportunity forinstructional design within a curricular context.Representation and Instructional Design in Computational Media: A SampleAnalysis from CalculusTo illustrate the kinds of issues that arise when one approaches representational issues from aninstructional design perspective, I will provide an example interspersed with discussion of basicissues raised in the various papers.Anchoring Student Learning in Concrete, Experientially Real Data: TheInadequacy of the “Big Three” Linked RepresentationsHot, bi-directional links between pairs of the traditional “Big Three” (numerical, graphical andcharacter-string) notations have dominated the attention of educators and researchers (see the paperby Even), and, indeed, have driven computer software design and calculator design in recentyears. This has had the effect of enlarging the representational island and increasing students’ability to move around on it, but it has failed to connect to the students’ experiential mainland. Inthe words of Anna Sfard in her plenary address (with Patrick Thompson) at the PME-NA AnnualMeeting in October, 1994, “If the representations only represent each other, then the emperor isonly clothes.” A strong illustration of the kinds of learning that take place in the absence ofanchoring in concrete experience is given by Schoenfeld, et al. (1994). These researchersprovided an extremely detailed analysis of a single student’s learning in a linked “Big Three”environment, supplemented by a version of a target game based on Green Globs which involvesstudents defining functions whose graphs will pass through as many pre-given, randomlygenerated points on the plane as possible. The student’s learning was characterized by fragility,instability and disconnectedness from other knowledge or sense-making capability. (This was acapable, mature college-age student who had taken several prior courses involving algebra.) Thestudent’s conception of function was “only clothes” in the sense that, for her, the representationswere only referring to one another and not to any data associated with phenomena or situationsgrounded in her experience. By contrast, for the researchers the representations were physicalembodiments of their own ideas of function. Other reports of student learning difficulty in multiplelinked function environments indicate the inadequacy of linked representations and the strong needto provide experiential anchors for function representations.Our current work in the SimCalc Project (Kaput, Roschelle & Stroup, in press) puts phenomenaand situations at the center and treats the various representations of functions as means forunderstanding and reasoning about those phenomena and situations as reflected in Figure 1.----
Page 8Figure 1. Putting Phenomena at the CenterHere we are explicit about physical and cybernetic phenomena experienced as separate from theperson, but, perhaps generated in interaction with the person. Of course there are other kinds ofphenomena of direct interest, such as notational phenomena - the behavior of notations as weinteract with them. Or experienced kinesthetic phenomena, as occur in whole body motion orwhen a student moves an object using her hand.The reader will notice the bidirectionality of the arrows in Figure 1. The next two sections discussinterpretations of these arrows.From Representing to Creating and Controlling Situations or PhenomenaHistorically, we have assumed that mathematics was to be used to represent aspects of situations orphenomena within the notational systems of mathematics (often referred to as “modeling”) in orderto reason about and make sense of those situations or phenomena, which were taken as given.This can be taken as our intended meaning of the outward-pointing arrows. A secondinterpretation of the outward pointing arrows is data-transfer, from the physical environment to acomputational one through the use of measuring devices connected to a computational devicewhich can display the data in some mathematical notation system. Our work has focused on themathematics of motion, so in many cases, the notations describe velocities, positions, times, andcombinations of these in graphs, equations and tables, with our emphasis on graphs indicated inthe figure. However, such motion phenomena, and others such as fluid-flow, are not onlymodeled by the notations that describe them, they can be controlled by those notations. Thesephenomena can either be cybernetic, as with screen-objects whose movement is controlled by----
Page 9mathematical functions - represented as graphs, equations or tables - or physical, as with toy carslinked to a computer where their motion is controlled by mathematical functions defined on thecomputer. This can be taken as our intended meaning of the inward-pointing arrows. See(Nemirovsky, Kaput & Roschelle, 1998) for more details and concrete examples.These kinds of affordances turn a fundamental representational relationship between mathematicsand experience from one-way to bi-directional. This in turn supports a much tighter and morerapid interaction on which to base learning. Because the mathematical notation that controls aphenomenon also can be treated as a model of it, one can test a hypothetical model againstexpectations or predictions immediately - by “running” it. Note that the feedback structure oftenrequires twoFigure 2: Matching Red’s Position by Controlling Blue’s Velocityphenomena, P and P’, where P is given as a target and P’ is defined or controlled by the studentattempting to match P in another system of description. In cases involving rate-totals connections,P may act as a referent phenomenon for either a rate or totals description, and the student inputsone of these and gets feedback in terms of the other. For example, (See Figure 2) suppose we aregiven a vertical motion P of a Red elevator (on the left) and its position description (a position vs.time graph, for instance as in Figure 2). Then the student controls the motion P’ of a parallel Blueelevator (on Red’s right) by constructing its velocity, say a graph (or even a formula) to match themotion P of Red with feedback available by watching both elevators run simultaneously. Ofcentral importance is that the student’s intentions can be made visible, explicit and testable throughthe phenomena that the student controls. By exercising our many representational options, we canaddress an enormous range of learning objectives.From Multiple Linked Representations to Multiple Linked Descriptions ofExperientially Real Situations or PhenomenaFirst we set the stage for the distinction between different representations of the “same” descriptionand the “same” representation of “different” descriptions. Situations or phenomena admitting ofquantitative analysis almost always have two kinds of quantitative descriptions, one describing thetotal amount of the quantity at hand with respect to some other quantity such as time, and the other----
Page 10describing its rate of change with respect to that other quantity. In the SimCalc Project we havetaken the perspective that understanding the two-way relations between totals and rates descriptionsof varying quantities (and the situations that they describe), is a fundamental aspect of quantitativereasoning. It is exactly this relationship that is at the heart of the Fundamental Theorem ofCalculus, and indeed, at the heart of Calculus itself. Given the centrality of Calculus to ourcurriculum and to the mathematics, science and technology of western civilization, the connectionsbetween these two types of description is difficult to overestimate.Now, these rates-totals connections can be instantiated in any of the different representationsystems we have mentioned so far because both the rates and the totals descriptions are, in mostcases of interest to us, functions. In Figure 2 we saw a case where the two descriptions were bothinstantiated in the same representation system, coordinate graphs, but mentioned that they couldhave been represented across different systems. Furthermore, we continue to take advantage oflinked representations, so that we not only can connect graphs and formulas, we can cross-connect, for example, a rate graph to a totals formula. See Figure 3 for an illustration of the manypossible linkages.Figure 3: Linked Representations AND Linked DescriptionsSimilar points could be made about different descriptions in other mathematical domains, such asstatistical data analysis, probability, graph theory (especially rich representationally). In all thesecases, differences at the level of descriptions runs structurally deeper than do the differences inrepresentation systems, although it is of course the case that representations differ in their abilitiesto render that structure available to us. This description-representation issue is worthy of muchfurther study, especially cross-domain study.----
Page 11This new description-representation distinction adds another burden to our terminology and theneed to be explicit. To speak only of a “model” or a “representation” without further elaboration isclearly inadequate given the many different interpretations possible.The Invisible Effects of Static, Inert Media: Dominance of Character StringNotation SchemesFish don’t know they are wet. Similarly, we have historically taken without question the static,inert media in which notation schemes have been instantiated and used in representational andcomputational ways. And just as fish have gills and fins rather than lungs and legs, our ways ofdoing mathematics have certain features that evolved due to the nature of the media in which theyevolved. Thus, among notation systems, one type, the character string based systems, dominatesthe others. Character string systems are very compact, they can support intricate syntax for bothreading and transformations and so can support complex computation and reasoning far beyondwhat could be achieved by a notationally naked mind, and they can be representationally neutral inthe sense of being able to denote enormous varieties of referents without needing to share thevisual features of what they are representing (e.g., the meaning of “big” does not require the word“big” to be big).Changes from static inert to dynamic interactive medium affect the place of character string notationsystems in mathematics in at least three basic ways. The first is by enabling character strings to“come alive” - to embody procedures, or algorithms, that can be autonomously executed in order to“do something,” carry out a computation, evaluate an expression, perform a translation ortransformation, etc. - essentially to do anything a computer program can do. Second, the mediumcan support live linkages among notation systems. Third, it can support the instantiation of newnotation schemes that can be linked with others, new or old. We now turn attention to this third,most promising affordance.This deep notational bias is reflected in a deep cultural and curricular bias that has led to the bulk ofschool mathematics being centered on the teaching of reading and writing in character-basednotation schemes, especially routines for calculation or reasdoning. The momentum of thisnotational bias from static inert media led to the design of most first generation electroniceducational technology in mathematics to require character string input, e.g., keyboards andcalculator keys. This character string orientation extends to underlying issues of legitimacy - whatcounts as significant mathematics is typically taken to be mathematics expressed in terms ofcharacter strings. While we have no reason to expect that the extraordinarily powerful of characterstring mathematics will ever or should ever disappear or be displaced, we should expect that it willbe increasingly augmented by mathematics expressed in other notational styles, especially thosethat draw upon the new structurable flexibilities and visually rich notations of dynamic, graphicand direct-manipulation media. We see in the papers by Edwards, Even and Hitt (this issue),gradual movement away from exclusive dependence on character strings. Of course, the work byJanvier beginning in the 1970’s, pioneered the study of graphical notation systems. Hisposthumously published paper in this issue raises the question of how the subtle and pervasivesemantics of time structures descriptions of phenomena, whether or not the key variables used todescribe the phenomena are themselves temporal.Historical Approaches: Globally Defined Functions Are PrimaryHistorical necessity pushed mathematics towards globally defined functions - a character stringdefinition of a function or quantitative relationship was globally defined (over its natural domain)essentially by default - through the identification of the function or quantitative relationship with thecharacter string. But, of course, the phenomena and situations that these are used to model do notusually have such “simply” definable characteristics over unbounded domains. Rather, they are----
Page 12defined over finite extent (domains) and often change their essential character within that finiteextent. Much elaborate and beautiful mathematics has been produced to achieve the flexibility thatclosed form functions do not admit - infinite series approximations of various kinds, Fourierseries, and so on - all of which require an apprenticeship in the algebraic world that most citizenstoday cannot afford and whose role in sense-making is fundamentally transformed by newtechnologies. (Note that their significance does not disappear, but is played out in different ways,e.g., by being embodied in tools that are used by people who may not need to master that style ofmathematics.)A Alternative: Piecewise Defined, Visually Editable FunctionsIn MathWorlds, an important feature is the ability to construct and modify functions that areconnected to motion and other simulations through strictly graphical means - direct manipulation ofpiecewise defined functions (see also Yerushalmy, 1997). As indicated in Figure 4 below, one cancreate a piecewise constant velocity function that controls the motion of the Clown in the motionsimulation by dragging the black dots associated with parts of the graphs.Drag this hotspotleft or right toadjust the initialtimeDrag this hotspotleft or right toadjust the initialtimeDrag this hotspotleft or right toadjust the initialtimeFigure 4: Visually Editable Velocity GraphsAn important point is that the historical power of algebra lay in its status as an action notationscheme, one that supports structured actions on well-formed character strings in support ofmathematical reasoning and modeling. By contrast, in static inert media, coordinate graphs havetypically been used as static notations - to be inspected, and analyzed as fixed inscriptions. Inrecent times, by instantiating coordinate graphs in dynamic interactive media as linked to algebraicrepresentations, as noted above they become manipulable, but in global form. That is, one actsuniformly on the function over its entire domain - translating, stretching, reflecting, etc. (SeeConfrey, 1991) However, the approach illustrated in Figure 4 goes a step beyond this, in notrequiring an algebraic interpretation or linkage. Instead, much in the spirit of Dynamic Geometry,the graphs are directly manipulable: pieces can be inserted and modified, etc. The manipulability ofalgebraic notation has been replaced by a new kind of manipulabilty.----
Page 13Three sets of reasons lie behind our use in the SimCalc Project of piecewise defined functions: (1)from a modeling orientation, phenomena occur in segments of time, each of which has boundedextent, so the phenomenon usually is not regarded as being defined in a unitary fashion “forever;”(2) work by Nemirovsky and colleagues has established that people naturally tend to interpretgraphs on the basis of interval-analyses (Nemirovsky, 1994, 1996; Nemirovsky, Tierney &Wright, in press; Tierney & Nemirovsky, 1995; Monk & Nemirovsky, 1994), whereincomplicated graphs linked to phenomena are parsed one section or interval at a time according tothe student’s current experience of both the graph and whatever it is linked to; and (3) in order toachieve the variation needed to avoid the logical and psychological degeneracy of constant or linearfunctions while preserving computational tractability, we chose to begin with graphically editablepiecewise constant and linear functions whose slope and area computations can be approachedusing simple arithmetic and geometry, temporarily avoiding the subtleties of limits andapproximations (Kaput, Roschelle & Stroup, in press).Curricular Implications: Moving Past the Algebra BottleneckIn addition to the factors identified above, the weight of the historical tradition based in theextraordinary achievements of algebra-based mathematics in the hands of the masters who inventedand used it (these constitute some of the greatest intellectual achievements of Western civilization),the assumption that the mathematics of change and variation, including the ideas and techniques ofcalculus, should be learned exclusively in the language of algebra - this assumption - hascontinued by default up to the current time. It is deeply embedded in our curriculum and ourinstitutional structures (Kaput, 1997), and indeed, it is reflected in both our popular and intellectualcultures.No one should presume to challenge the power of algebra, and indeed, much mathematicsabsolutely requires algebra, including most of classical mathematics. Rather, we can and shouldchallenge its currently dominant place in the curriculum as a prerequisite for access to otherimportant mathematics by students who will never need the specialized techniques that the algebramakes possible. On the other hand, all that we have learned about student learning suggests thatstudents bring enormously rich resources to us that are intimately tied to their natural ways of beingin and constructing their world. Our ongoing work provides strong evidence that these resources,combined with technologically supported learning environments that instantiate new notationsystems and ways of acting upon them, offer dramatically new possibilities for mathematicslearning.ConclusionsWe are entering a new era in the study of representation, where new issues need attention and newperspectives and analytical tools are becoming available. The same factors that are being felt acrossour society and our intellectual landscape are at work here: new technologies yield new media thatalter the semiotic foundations of mathematics; situationist perspectives provide new ways tointegrate, or supersede, traditional cognitivist and social-psychological perspective and methods;advances in the sociology of science provide fresh perspectives on the creation and communicationof the inscriptions that move among us and that constitute an essential dimension of our world; andnew obligations are opposed upon mathematics educators to render ever more mathematicslearnable by ever more, and hence more diverse, people.We are entering a new era in the study of representation, where new issues need attention and newperspectives and analytical tools are becoming available. The same factors that are being felt acrossour society and our intellectual landscape are at work here: new technologies yield new media thatalter the semiotic foundations of mathematics; situationist perspectives provide new ways tointegrate, or supercede, traditional cognitivist and social-psychological perspectives and methods;advances in the sociology of science provide fresh perspectives on the creation and communication----
Page 14of the inscriptions that move among us and constitute an essential dimension of our world; andnew obligations are imposed upon mathematics educators to render ever more mathematicslearnable by ever more, and hence more diverse, people.I hope that this paper, as a small twist of the representational kaleidoscope, has provided a fewindications of how the work in this collection can serve the needs of the new era. Of specialinterest is how we may extend this work to inform the design of instruction, especially in thecontexts of school-implementable curricula, which is where we ultimately turn for a fair test of anyof our endeavors.----
Page 15ReferencesAnderson, J. R., Reder, L. M., & Simon, H. A. (1996). Situated learning and education.Educational Researcher,25(4), 5-11.Anderson, J. R., Reder, L. M., & Simon, H. A. (1997). Situative vs. cognitive perspective:Form vs. substance. Educational Researcher,26(1), 18-21.Cobb, P., & Bowers, J. (in press). Cognitive and situated learning perspectives in theory andpractice. To appear in Educational Researcher.Cobb, P., Yackel, E., & McClain, K. (in press). Symbolizing and communicating inmathematics classrooms. Hillsdale, NJ: Erlbaum.Confrey, J., & Smith, E. (1991). A framework for functions: Prototypes, multiplerepresentations, and transformations. Paper presented at the Proceedings of the thirteenth annualmeeting of Psychology of Mathematics Education-NA, Blacksburg, VA.diSessa, A. A., Hammer, D., Sherin, B. & Kolpakowski, T. (1991). Inventing graphing: Meta-representational expertise in children. Journal of Mathematical Behavior, 10, 117-160.Donald, M. (1991). Origins of the modern mind: Three stages in the evolution of culture andcognition. Cambridge, MA: Harvard University Press.Goodman, N. (1976). Languages of art. (Revised ed.). Amherst, MA: University ofMassachusetts Press.Goldin, G. A., & Kaput, J. (1992). Comments at Working Group on Representations, AnnualMeeting of the International Group for the Psychology of Mathematics Education, Durham, NewHampshire.Gravemeijer, K., Cobb, P., Bowers, J., & Whitenack, J. (in press). Symbolizing, modeling,and instructional design. To appear in P. Cobb, E. Yackel, and K. McClain (Eds.), Symbolizingand communicating in mathematics classrooms. Hillsdale, NJ: Erlbaum.Greeno, J. (1997). On claims that answer the wrong questions. Educational Researcher, 26(1),5-17.Hoyles, C., & Noss, R. (Eds.) (1998). Mathematics for a new millennium. London, England:Springer-Verlag.Kaput, J. (1985). Representation and problem solving: Methodological issues related tomodeling. In E. Silver (Ed.), Teaching and learning mathematical problem solving: Multipleresearch perspectives, (pp. 381-398). Hillsdale, NJ: Lawrence Erlbaum.Kaput, J. (1987). Toward a theory of symbol use in mathematics. In C. Janvier (Ed.), Problemsof representation in mathematics learning and problem solving, (pp. 159-196). Hillsdale, NJ:Lawrence Erlbaum Associates.Kaput, J. (1991). Notations and representations as mediators of constructive processes. In E. v.Glasersfeld (Ed.), Constructivism and mathematics education, (pp. 53-74). Dordrecht,Netherlands: Kluwer.----
Page 16Kaput, J. (1997). Rethinking calculus in terms of learning and thinking. The AmericanMathematics Monthly, 104 (8), 731-737.Kaput, J., Roschelle, J., & Stroup, W. (in press). SimCalc: Accelerating Students' Engagementwith the Math of Change. To appear in M. Jacobson & R. Kozma (Eds.) Learning the sciences ofthe 21
st
century: Research, design, and implementation of advanced technology learningenvironments. Hillsdale, NJ: Lawrence Erlbaum.Latour, B. (1987). Science in action: How to follow scientists and engineers through society.Cambridge, MA: Harvard University Press.Latour, B. (1993). La clef de Berlin et autres lecons d’un amateur de sciences [The key to Berlinand other lessons of a science lover]. Paris: Editions la Decouverte.Meira, L. (1992). The microevolution of mathematical representations in children's activity.Paper presented at the Proceedings of the Sixteenth Annual Conference of the PME, Vol. 2,Durham, NH.Monk, S., & Nemirovsky, R. (1994) The Case of Dan: Student construction of a functionalsituation through visual attributes. In E. Dubinsky, J. Kaput, & A. Schoenfeld (Eds.), Researchin Collegiate Mathematics Education. Vol. 1. Providence, RI: American Mathematics Society,139-168.Nemirovsky, R. (1994). On ways of symbolizing: The case of Laura and velocity sign. TheJournal of Mathematical Behavior, 13, 389-422Nemirovsky, R. (1996). Mathematical narratives. In N. Bednarz, C. Kieran, & L. Lee (Eds.),Approaches to algebra: Perspectives for research and teaching, (p. 197-223). Dordrecht, TheNetherlands: Kluwer Academic Publishers.Nemirovsky, R., Kaput, J., & Roschelle, J. (1998). Enlarging mathematical activity frommodeling phenomena to generating phenomena. Paper presented at the Proceedings of the 22
nd
International Conference of the Psychology of Mathematics Education in Stellenbosch, SouthAfrica.Nemirovsky, R., & Monk, S. (in press). “If you look at it the other way...” An exploration intothe nature of symbolizing. In Cobb, P., Yackel, E. and McClain, K. (Eds.) Symbolizing andcommunicating in mathematics classrooms. Hillsdale, NJ: Erlbaum.Nemirovsky, R., Tierney, C., Wright, T. (in press). Body motion and graphing. Cognition andInstruction.Palmer, S. E. (1977). Fundamental aspects of cognitive representation. In E. Rosch & B. B.Lloyd (Eds.), Cognition and categorization. Hillsdale, NJ: Lawrence Erlbaum Associates.Roth, W-M., & McGinn, M. K. (1998). Inscriptions: Toward a theory of representing as socialpractice. Review of Education Research, 68(1), 35-59.Schoenfeld, A., Smith, J., & Arcavi, A. (1994). Learning: The microgenetic analysis of onestudent’s evolving understanding of a complex subject matter domain. In R. Glaser (Ed.),Advances in instructional psychology, (Vol. 4). Hillsdale, NJ: Lawrence Erlbaum.Shaffer, D., & Kaput, J. (submitted). Mathematics and virtual culture: An evolutionaryperspective on technology and mathematics education. Educational Studies in Mathematics.----
Page 17Tierney, C., & Nemirovsky, R. (1995). Children's Graphing of Changing Situations. Presentedat the 1995 Annual Meeting of the American Educational Research Association. San Francisco,CA.Werner, H., & Kaplan, B. (1962). Symbol formation. New York: Wiley.Yerushalmy, M. (1997). Emergence of new schemes for solving algebra word problems: Theimpact of technology and the function approach. Paper presented at the Proceedings of the 21
st
Conference of the International Group for the Psychology of Mathematics Education, Lahti,Finland.
.